The Stefan-Boltzmann law describes how much thermal energy any object radiates purely because of its temperature — no contact, no medium, and no wind required. It's the physics behind why the Sun heats the Earth across empty space, why a hot stovetop element glows and radiates heat you can feel without touching it, and why spacecraft need radiators to shed waste heat in the vacuum of space.
How the Stefan-Boltzmann law works
The law states that radiated power P equals ε·σ·A·T⁴, where σ is the Stefan-Boltzmann constant (5.670374419×10⁻⁸ W/(m²·K⁴)) — a fixed physical constant derived from more fundamental constants (Planck's constant, the speed of light, and Boltzmann's constant) via integrating Planck's law of blackbody radiation over all wavelengths. The formula assumes the object radiates into surroundings at a much lower temperature; for precise heat-exchange calculations between two radiating bodies, engineers typically use the net form P = εσA(T₁⁴ − T₂⁴).
The single most important feature of the law is the fourth-power dependence on temperature. A linear relationship would mean doubling temperature doubles radiated power; the actual T⁴ relationship means doubling temperature multiplies power by 16. This is why very hot objects — a lightbulb filament at ~2,800 K, or the Sun's photosphere at ~5,778 K — radiate disproportionately more energy than modestly warm ones, and why thermal engineering at high temperatures is dominated by radiative, not convective, heat loss.
Inputs and what they mean
Temperature (T) must be in kelvin. This is not optional: Celsius and Fahrenheit have arbitrary zero points, so raising a Celsius reading to the fourth power gives a meaningless result. Convert first (K = °C + 273.15) if your data is in another scale.
Surface area (A) is the total area actually exposed to radiate — for irregular shapes, this can be harder to estimate than volume or mass, and it's a common source of error in real engineering calculations.
Emissivity (ε) ranges from 0 (a perfect reflector, radiates nothing) to 1 (an ideal blackbody). It depends on the material, surface finish, and to a lesser extent wavelength and viewing angle. Polished metals (aluminum foil, chrome) have emissivity as low as 0.02–0.1; most non-metals, paints, and biological surfaces (including human skin) sit at 0.9–0.98 regardless of visible color, because emissivity is mostly a function of infrared, not visible-light, reflectance.
Limits and edge cases
This calculator computes the power radiated by an object into surroundings effectively at absolute zero — it does not subtract incoming radiation from the environment. For net heat exchange between two bodies at different temperatures, subtract each body's radiated power from the other's, or use the two-temperature form P = εσA(T_hot⁴ − T_cold⁴).
The formula also assumes emissivity is constant across all relevant wavelengths (a 'grey body' approximation). Real materials can have emissivity that varies significantly with wavelength or temperature, which matters for precise infrared instrumentation or high-temperature furnace design — in those cases, consult material-specific emissivity tables or a wavelength-resolved (spectral) emissivity model rather than this single-value approximation.